Talk:5-manifolds: 1-connected

1 Conjecture about mapping class groups of 1-connected 5-manifolds

Let $M$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}M$ be a closed, smooth, 1-connected 5-manifold, Matthias Kreck and Diarmuid Crowley conjecture that there is an isomorphism of abelian groups

$\displaystyle \pi_0(\SDiff(M)) \cong \Omega_6(B^2(M))$ $$\displaystyle \pi_0(\SDiff(M)) \cong \Omega_6(B^2(M))$$

where $B^2(M)$$B^2(M)$ is the normal $2$$2$-type of $M$$M$ as defined in [Kreck1999]. For example, if $M$$M$ is Spinable with $H_2(M) \cong H$$H_2(M) \cong H$ then

$\displaystyle \Omega_6(B^2(M)) \cong \Omega_6^{\Spin}(K(H, 2)).$ $$\displaystyle \Omega_6(B^2(M)) \cong \Omega_6^{\Spin}(K(H, 2)).$$

At present we are checking the details of the proof of this conjecture using the methods of [Kreck1999].

Diarmuid Crowley 10:02, 29 September 2009 (UTC)

2 Earlier work of Fang

The group $\pi_0\SDiff(M)$$\pi_0\SDiff(M)$ was computed in [Fang1993] provided that $H_2(M)$$H_2(M)$ has no $2$$2$-torsion and no $3$$3$-torsion.

Up-date of conjecture: module structure

If the conjecture above holds, then from the short exact sequence

$\displaystyle 0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\text{Diff}_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0 \quad (\ast)$ $$\displaystyle 0 \rightarrow \pi_0\SDiff(M) \rightarrow \pi_0\text{Diff}_+(M) \rightarrow \Aut(H_2(M)) \rightarrow 0 \quad (\ast)$$

we obtain an action of $\Aut(H_2(M))$$\Aut(H_2(M))$ on the abelian group $\pi_0\SDiff(M) \cong \Omega_6(B^2(M))$$\pi_0\SDiff(M) \cong \Omega_6(B^2(M))$. Diarmuid Crowley and Matthias Kreck also conjecture that the action of $\Aut(H_2(M))$$\Aut(H_2(M))$ is via the induced action on $B^2(M)$$B^2(M)$. In particular, if $M$$M$ is spinable with $H = H_2(M)$$H = H_2(M)$, then $\Aut(H_2(M))$$\Aut(H_2(M))$ acts on $K(H, 2)$$K(H, 2)$ in the obvious way and so on $\Omega_6^{\Spin}(K(H, 2))$$\Omega_6^{\Spin}(K(H, 2))$.